Scattering Control for the Wave Equation with Unknown Wave Speed

0.92 MB·

by  Peter Caday

Scattering Control for the Wave Equation with Unknown
Wave Speed
PeterCaday*, MaartenV.deHoop†, VitalyKatsnelson‡, andGuntherUhlmannk
January5,2017
7
1
0 Abstract
2
Consider the acoustic wave equation with unknown, not necessarily smooth, wavespeed
n
a c. Weproposeandstudyaniterativecontrolprocedurethaterasesthehistoryofawavefield
J up to a given depth in a medium, without any knowledge of c. In the context of seismic or

4 ultrasoundimaging,thiscanbeviewedasremovingmultiplereflectionsfromnormal-directed

wavefronts.
]
P
A
1 Introduction
.
h
t
a Considertheacousticwaveequationwithanunknownwavespeedc,notnecessarilysmooth,ona
m
finiteorinfinitedomainΩ Rn. AssumethatwecanprobeourdomainΩwitharbitraryCauchy
[ Ă
data outside of Ω, and measure the reflected waves outside Ω for sufficiently large time. The

1
inverse problem is to deduce c from these reflection data, and this is the basis for many wave-
v
0 basedimagingmethods,includingseismicandultrasoundimaging.
7 Toward this goal, we will define and study a time-reversal type iterative process, the scat-
0
1 tering control series. We were inspired by the work of Rose [14] in one dimension, who devel-
0 oped a “single-sided autofocusing” procedure and identified it as Volterra iteration for the clas-
.
1 sical Marchenko equation. The Marchenko equation solves the inverse problem for the one-
0 dimensional acoustic wave equation1, recovering c on a half-line from measurements made on
7
1 the boundary. In the course of our research, it became evident that the new procedure is quite
:
v closely linked to boundary control problems [2, 8], and has similar properties to Bingham et al.’s
i iterativetime-reversalcontrolprocedure[3].
X
Inessence,scatteringcontrolallowsustoisolatethedeepestportionofawavefieldgenerated
r
a
by given Cauchy data— behavior we demonstrate with both an exact and microlocal (asymp-
totically high-frequency) analysis. Along the way we present several applications of scattering
control, including the removal of multiple reflections and the measurement of energy content of
a wave field at a particular depth in Ω. In a future paper, we anticipate illustrating how to locate
discontinuitiesincandrecovercitself.
Inthemathematicalliterature,theinverseproblem’sdataaretypicallygivenontheboundary
ofΩ,intermsoftheDirichlet-to-Neumannmaporitsinverse. WefindthattheCauchydata-based
*[email protected] †[email protected] ‡[email protected] [email protected]
*†‡DepartmentofComputationalandAppliedMathematics,RiceUniversity.
kDepartmentofMathematics,UniversityofWashington.
1In fact, the Marchenko equation treats the constant-speed wave equation with potential, to which the one-
dimensionalacousticwaveequationcanbereducedbychangingcoordinates.
1
t
2T
T
x x
0
(a)WavefieldgeneratedbyCauchydatah0 (b) Wave field with trailing pulse added to
initialdata)
Figure1.1:(a)AdomainΩ(shaded)withunknownwavespeedcisprobedbyexteriorCauchydatah0.
Twodiscontinuitiesinc(dashed)scattertheincomingwave. (b)Anappropriatetrailingpulseadded
toh suppressesmultiplereflections.
0
reflectionmapallowsusamuchcleaneranalysis. Itisnothardtosee(cf.Proposition2.6)thatthe
Dirichlet-to-Neumann map determines the Cauchy data reflection map, so no extra information
isneeded.
We start with an informal, graphical introduction to the problem. Section 2 defines the scat-
tering control series rigorously and provides an exact analysis of its behavior and convergence
properties. Section3pursuesthesamequestionsfromamicrolocalperspective. Thediscrepancy
that arises between the exact and microlocal analyses allows us to provide more insight on con-
vergenceinSection4. Section5concludesbyconnectingourworktothatofRoseandMarchenko.
1.1 Motivation
Before defining the scattering control equation and series, we begin by motivating our problem
withagraphicalexample. InFigure1.1,thedomainisΩ x 0 R,withapiecewiseconstant
“ t ą u Ă
wave speed c having two discontinuities. We extend c to all of R, but assume it is known only
outside Ω. Now consider the solution of the acoustic wave equation on R for time t 0,2T ,
P r s
withrightward-travelingCauchydatah supportedoutsideΩ. Theinitialwavescattersfromthe
0
discontinuitiesinc,producinganinfinitesequenceofreflections(Figure1.1(a)).
In imaging, one attempts to recover c or some proxy for it. In many imaging algorithms cur-
rently in use, only waves having undergone a single reflection (so-called primary reflections) are
typicallydesired,whiletheremainingmultiplereflectionsonlycomplicatetheinterpretationofthe
data. Asaresult,muchresearchinseismicimaginghasbeendirectedtowardremovingoratten-
uatingmultiplereflections.
For the problem at hand, it is plausible (and can be proven) that by adding a proper control,
or trailing pulse to the initial data, the multiple reflections may be suppressed, at the cost of a
harmless additional outgoing pulse (Figure 1.1(b)). If c were known inside the domain (cf. §3.4),
an appropriate control may be constructed microlocally under some geometric conditions. The
issue,ofcourse,istofindthecontrolknowingonlythereflectionresponseofΩ.
Rather than attacking the multiple reflection suppression problem, however, we consider a
2
relatedproblemobtainedbyfocusingontheinterior,ratherthanexterior,ofΩ. ReturningtoFig-
ure (b), we note that the wave field rightmost portion of the medium contains a single, purely
transmitted wave, which we call the direct transmission of the initial data h0. Slightly more pre-
cisely,thewavefieldinsideΩattime2T isgeneratedexactlybythedirecttransmissionattimeT.
Thecontrolhasthereforeisolatedthedirecttransmission;ourproblemistofindsuchacontrolfor
agivenh usingonlyinformationavailableoutsideΩ.
0
1.2 Almostdirecttransmission
At its heart, the direct transmission is a geometric optics construction, and is valid only in the
high-frequency limit where geometric optics holds. Consequently, the directly transmitted wave
fieldcanbeisolatedonlymicrolocally(modulosmoothfunctions). Wewillconsiderthegeometric
optics viewpoint later, but initially avoid a microlocal approach, as follows. Informally, suppose
h createsawavethatentersΩattime0,travellingnormaltotheboundary. AtalatertimeT,the
0
directly transmitted wave may be singled out from all others by its distance from the boundary:
namely, T (as long as it has not crossed the cut locus). Here, distance is the travel time distance,
whichforcsmoothisRiemanniandistanceinthemetricc 2dx2.
´
Withthisinmind,givenCauchydatah supportedjustoutsideΩwesubstituteforthedirect
0
transmission the almost direct transmission, the part of the wave field of h0 at time T of depth at
leastT. Moreprecisely,letΘbeadomaincontainingΩandsupph ;thenletΘ Θbethesetof
0 T
Ă
points in Θ greater than distance T from the boundary. The almost direct transmission of initial
datah attimeT istherestrictiontoΘ ofitswavefieldatt T (Figure1.2).
0 T
“
h
0 Support of wave field at time T
∂Θ
∂Ω Almost direct transmission
T T T Directly transmitted ray
Scattered rays
∂Θ
T
Figure1.2: Almostdirecttransmissionofinitialdatah0attimeT 0.
ą
The nonzero volume of Θ Ω means that some multiply reflected rays may still reach Θ .
T
z
Hence, we have in mind taking a limit as Θ Ω and the support of h approaches a point on
0
Ñ
Ω. Inthislimit,thesupportofthealmostdirecttransmissionconvergestoapointalongthenor-
B
maldirectly-transmittedray,forsufficientlysmallT (atleastintheabsenceofcausticsandbefore
reachingthecutlocus);seeFigure1.3.
2 Exact scattering control
We set up the problem and our notation in 2.1, then introduce the scattering control procedure
in 2.2, where we study its behavior and convergence properties. The final result, expressed in
Corollary2.4,isthatscatteringcontrolrecoversthealmostdirecttransmission’swavefieldoutside
Θ, modulo harmonic extensions. In 2.3, we apply this to recover the energy (with a harmonic
3
h ∂Θ
0
∂Ω
∂Θ
T
Figure 1.3: Shrinking the support of the initial data h0 to a point. The dashed line indicates the nor-
mal geodesic from that point; the support of the almost direct transmission shrinks to a point on the
geodesic.
extension) and kinetic energy of this portion of the wave field. Proofs for the results in these
sectionsfollowin2.4.
2.1 Setup
2.1.1 Uniquecontinuation
LetΩ Rn beaLipschitzdomain,andasoundspeedcwithc,c 1 L Rn .
´ 8
Ď P p q
Thesolerestrictionweimposeoncisthatitsatisfyacertainformofuniquecontinuation. More
precisely, assume there is a Lipschitz distance function d x,y such that any u C R,H1 Rn
p q P p p qq
satisfyingeither:
• u, u 0fort 0andd x,x T (finitespeedofpropagation)
t 0
B “ “ p q ă
• u 0onaneighborhoodof T,T x (uniquecontinuation)
0
“ r´ sˆt u
isalsozeroonthelightdiamond
D x ,T t,x d x,x T |t| ,
0 0
p q “ tp q | p q ă ´ u
if 2 c2∆ u 0onaneighborhoodofD x ,T ,foranyx Rn,T 0.
t 0 0
pB ´ q “ p q P ą
While the set of wavespeeds with this property has not been settled in general, several large
classes of c are eligible, stemming from the well-known work of Tataru [21]. Originally known
forsmoothsoundspeeds[16,Theorem4],StefanovandUhlmannlaterextendedthistopiecewise
smooth speeds with conormal singularities [17, Theorem 6.1], and Kirpichnikova and Kurylev
to a class of piecewise smooth speeds in a certain kind of polyhedral domain [11, §5.1]. The
correspondingtraveltimed x,y istheinfimumofthelengthsofallC1 curvesγ s connectingx
p q p q
andy,measuredinthemetricc 2dx2,suchthatγ 1 singsuppc hasmeasurezero.
´ ´
p q
2.1.2 Geometricsetup
Next, let us set up the geometry of our problem. We will probe Ω with Cauchy data (an initial
pulse) concentrated close to Ω, in some Lipschitz domain Θ Ω. We will add to this initial pulse
Ą
a Cauchy data control (a tail) supported outside Θ, whose role is to remove multiple reflections
up to a certain depth, controlled by a time parameter T 0, 1 diamΩ . This will require us to
P p 2 q
considercontrolssupportedinaLipschitzneighborhoodΥofΘthatsatisfiesd Υ,Θ 2T and
pB q ą
isotherwisearbitrary.
4
WhileweareinterestedinwhatoccursinsideΩ,theinitialpulseregionΘwillactuallyplaya
largerroleintheanalysis. First,definethedepthd x ofapointxinsideΘ:
˚Θp q
d x, Θ , x Θ,
d x ` p B q P (2.1)
˚Θ
p q “ # d x, Θ , x Θ.
´ p B q R
Largervaluesofd arethereforedeeperinsideΘ. Foreacht,define2 theopensets
˚Θ
Θ x Υ d x t ,
t ˚Θ
“ t P | p q ą u (2.2)
Θ x Υ d x t .
‹t ˚Θ
“ t P | p q ă u
Asin(2.2)above,weuseasuperscript toindicatesetsandfunctionspaceslyingoutside,rather
‹
thaninside,someregion.
2.1.3 Acousticwaveequation
LetC˜ bethespaceofCauchydataofinterest:
C˜ H1 Υ L2 Υ , (2.3)
0
“ p q‘ p q
consideredasaHilbertspacewiththeenergyinnerproduct
f ,f , g ,g ∇f x ∇g x c 2f x g x dx. (2.4)
0 1 0 1 0 0 ´ 1 1
p q p q “ p q¨ p q` p q p q
Υ
ż
WithinC˜ defineth@esubspacesofCDauchy`datasupportedinsideandoutsid˘eΘ :
t
H H1 Θ L2 Θ , H H ,
t 0 t t 0
“ p q‘ p q “ (2.5)
H˜ H1 Θ L2 Θ , H˜ H˜ .
‹t 0 ‹t ‹t ‹ ‹0
“ p q‘ p q “
DefinetheenergyandkineticenergyofCauchydatah h ,h C˜ inasubsetW Rn:
0 1
“ p q P Ď
E h |∇h |2 c 2|h |2 dx, KE h c 2|h |2 dx. (2.6)
W 0 ´ 1 W ´ 1
p q “ ` p q “
W W
ż ´ ¯ ż
Next,defineF tobethesolutionoperator[13]fortheacousticwaveinitialvalueproblem:
2 c2∆ u 0,
t
pB ´ q “
F: H1pRnq‘L2pRnq Ñ CpR,H1pRnqq, Fph0,h1q “ u s.t. $ u(cid:12)(cid:12)t 0 “ h0, (2.7)
’ (cid:12) “
& Btu(cid:12)t 0 “ h1.
“
LetRs propagateCauchydataattimet 0toCauchydataatt s:’%
“ “
R F, F : H1 Rn L2 Rn H1 Rn L2 Rn . (2.8)
s t
“ p B q t s p q‘ p q Ñ p q‘ p q
ˇ “
NowcombineR withatime-reveˇrsaloperatorν: C˜ C˜,definingforagivenT
s ˇ Ñ
R ν R , ν: f ,f f , f . (2.9)
2T 0 1 0 1
“ ˝ p q ÞÑ p ´ q
Inourproblem,onlywavesinteractingwith Ω,c intime2T areofinterest. Consequently,letus
p q
ignoreCauchydatanotinteractingwithΘ,asfollows.
Let G H˜ R H1 Rn Θ L2 Rn Θ be the space of Cauchy data in C˜ whose wave
‹ 2T 0
fieldsvani“shonXΘatt p 0apndzt q‘2T. LpetCz bqeqitsorthogonal complementinsideC˜, andH its
orthogonalcomple`men“tinsideH˜“. Withthisdefi˘nition,R mapsCtoitselfisometrically. ‹t
‹t 2T
2WetacitlyassumethroughoutthatΘt,Θ‹t areLipschitz.
5
2.1.4 ProjectionsinsideandoutsideΘt
The final ingredients needed for the iterative scheme are restrictions of Cauchy data inside and
outside Θ. While a hard cutoff is natural, it is not a bounded operator in energy space: a jump
at Θ will have infinite energy. The natural replacements are Hilbert space projections. More
B
generally,weconsiderprojectionsinsideandoutsideΘ .
t
Letπ ,π betheorthogonalprojectionsofContoH ,H respectively;letπ 1 π . Asusual,
t t‹ t ‹t t t
“ ´
write π π , π π . The complementary projection I π π is the orthogonal projection
0 ‹ 0‹ t t‹
“ “ ´ ´
ontoI ,theorthogonalcomplementtoH H inC. Itmaybedescribedbythefollowinglemma,
t t ‹t
‘
whichisinessencetheDirichletprinciple.
Lemma 2.1. It consists of all functions of the form i0,0 with i0 the harmonic extension of some g
p q P
H1{2 Θt toafunctioninH01 Υ .
pB q p q
Lemma2.1providestwousefulpiecesofinformation. First,I I isindependentofc. Secondly,
0
“
wecanidentifythebehavioroftheprojectionsπ ,π . InsideΘ theprojectionπ hequalsh,while
t t‹ t t
outside Θ , it agrees with the I component of h, which is the harmonic extension of h to Υ
t t |BΘt
(with zero trace on Υ). Similarly, π h is zero on Θ , and outside Θ equals h with this harmonic
t‹ t t
B
extensionsubtracted.
Itwillbeusefultohaveanameforthebehaviorofπth,andsowedefinethenotionofstationary
harmonicity:
Definition. Cauchy data h0,h1 are stationary harmonic on W Rn if h0 W is harmonic and
p q Ď |
h 0.
1 W
| “
2.2 Scatteringcontrol
Suppose we have Cauchy data h H. We can probe Ω with h and observe Rh outside Ω. In
0 0 0
P
particular,thereflecteddataπ Rcanbemeasured,andfromthesedata,wewouldliketoprocure
‹
information about c inside Ω. However, multiple scattering as waves travel into and out of Ω
makesπ Rh difficulttointerpret.
‹ 0
In this section, we construct a control in H that eliminates multiple scattering in the wave
‹
fieldofh0 uptoadepthT insideΘ. Morespecifically,considerthealmostdirecttransmissionofh0:
Definition. Thealmostdirecttransmissionofh0 P HattimeT istherestrictionRTh0|ΘT.
Ideally,wewouldliketorecover(indirectly)thisrestrictedwavefield. IfconsideredasCauchy
dataon theambientspaceΥ, the almostdirecttransmissionhas infiniteenergyingeneral dueto
the sharp cutoff at the boundary of Θ . As a workaround, consider the almost direct transmis-
T
sion’sminimal-energyextensiontoΥ. Thisinvolvesaharmonicextensionofthefirstcomponent
ofCauchydata:
Definition. Theharmonicalmostdirecttransmissionofh0 attimeT is
h h h ,T π R h . (2.10)
DT DT 0 T T 0
“ p q “
By Lemma 2.1, h is equal to R h inside Θ ; outside Θ , its first component is extended
DT T 0 T T
harmonicallyfrom Θ ,whilethesecondcomponentisextendedbyzero.
T
B
6
2.2.1 Scatteringcontrolseries
OurmajortoolisaNeumannseries,thescatteringcontrolseries
h 8 π Rπ R ih , (2.11)
‹ ‹ 0
8 “ p q
i 0
ÿ“
formallysolvingthescatteringcontrolequation
I π Rπ R h h . (2.12)
‹ ‹ 0
p ´ q 8 “
The series in general does not converge in C; but it does converge in an appropriate weighted
space, as we show in Theorem 2.3. Applying π to (2.11), we see that h consists of h plus a
0
8
controlinH . Ourfirsttheoremcharacterizesthebehavioroftheseries.
‹
Theorem2.2. Leth0 P HandT P p0, 21 diamΘq. Thenisolatingthedeepestpartofthewavefieldofh0 is
equivalenttosummingthescatteringcontrolseries:
I π‹Rπ‹R h h0 R TπR2Th hDT andh h0 H‹. (2.13)
p ´ q 8 “ ðñ ´ 8 “ 8 P `
Above,R TπR2Th mayalsobereplacedbyR sπT sRT sh foranys 0,T .
´ 8 ´ ´ ` 8 P r s
Such an h , if it exists, is unique in C. As for the harmonic extension in hDT, it is equal to πR2Th
8 8
outsideΘ:
hDT Θ‹ “ j0 Θ‹, where πR2Th8 “ pj0,j1q, (2.14)
ˇ ˇ
andisbounded: ˇ ˇ
E h Ckh k (2.15)
Θ‹Tp DTq ď 0
forsomeC C c,T independentofh0.
“ p q
Equation (2.13) tells us that the wave field created by h inside Θ at t 2T is entirely due to
8 “
the harmonic almost direct transmission at t T (Figure 2.1). More generally, the wave field of
“
h agrees with that of h on its domain of influence. This is not true of h ’s wave field, where
DT 0
8
otherwaves, includingmultiplereflections, willpollutethewavefieldattime2T. Itfollowsthat
the tail h h enters Ω and carries all of the scattered energy of h out with it. We will see this
0 0
8 ´
from an energy standpoint in section 2.3 and from a microlocal (geometric optics) standpoint in
section3.
ThequestionnowistostudywhethertheNeumannseries(2.11)convergesatall. Notethatif
the series fails to converge, Theorem 2.2 guarantees that no other finite energy control in H can
‹
isolatetheharmonicalmostdirecttransmissionofh .
0
Since R is an isometry and π a projection, we have kπ Rπ Rk 1. From our later spectral
‹ ‹ ‹
ď
characterization, we know that kπ Rhk khk, strictly, for all h H . This is also true for a
‹ ‹
ă P
completely trivial reason: we eliminated G when constructing C. What hinders convergence
is that khk kπ Rhk might be arbitrarily small; in other words, almost all the energy could be
‹
´
reflectedoffΘ.
Inthenexttheorem,weinvestigateconvergenceviathespectraltheorem. Itturnsoutthatthe
onlyproblemisoutsideΘ;insideΘthepartialsums’wavefieldsatt 2T doconverge,andtheir
“
energiesareinfactmonotonicallydecreasing. WewillalsodemonstratethattheNeumannseries
7
t ∂Θ
2T
equalto
wavefield
scatteredinitial ofhDT
pulseandcontrol
T h
DT
0
2T 0 T d (x)
− h h0 h0 ∗Θ
∞−
·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„c„„„„„„„o„„„„„„„‚ntr„„„o„„„„„„„„l„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„i¶nitialpulse
Figure2.1: Illustrationofthewavefieldgeneratedbyscatteringcontrol,asgivenbyTheorem2.2.
converges in H for a dense set of h , and identify a larger space in which the Neumann series
0
convergesforanyh .
0
For the statement of the theorem, define J to be the following space of Cauchy data, which,
roughlyspeaking,remainscompletelyinsideorcompletelyoutsideΘintime2T:
J H R H H R H . (2.16)
‹ ‹
“ X p q ‘ X p q
` ˘ ` ˘
Letχ: C JbetheorthogonalprojectionontoJ.
Ñ
Theorem2.3. Withh0,T asinTheorem2.2,definethepartialsums
k
h π Rπ R ih . (2.17)
k ‹ ‹ 0
“ p q
i 0
ÿ“
Thenthedeepestpartofthewavefieldcanbe(indirectly)recoveredfrom hk regardlessofconvergenceof
t u
thescatteringcontrolseries:
lim R πR h R χh h , kπRh k kh k. (2.18)
T 2T k T 0 DT k DT
k ´ “ “ Œ
Ñ8
Thesetofh0 forwhichthescatteringcontrolseriesconvergesinC,
Q h H I π Rπ R 1h C , (2.19)
0 ‹ ‹ ´ 0
“ P p ´ q P
(cid:32) ˇ (
is dense in H. For all h0 H, the partial suˇm tails hk h0 converge in a weighted space that can be
P ´
formallywrittenas
I
1 χ C, N πRπ π Rπ . (2.20)
‹ ‹
?I N2p ´ q “ `
´
As an immediate corollary of (2.18), we recover in the limit the wave field generated by the
harmonicalmostdirecttransmissionoutsideΘ,usingonlyobservabledata.
8
Corollary2.4. LetF t,x Fh t T,x betheharmonicalmostdirecttransmission’swavefield.
DT DT
p q “ p qp ´ q
Then
Fhk t,x Fπ‹R2Thk t 2T,x FDT t,x ask , (2.21)
p qp q´p qp ´ q Ñ p q Ñ 8
theconvergencebeingH1 inspace,uniformlyint.
Weendthissectionwithtwosmallpropositions. ThefirstcharacterizesthespaceH contain-
‹
ing the Cauchy data controls. Essentially, each control is supported in a 2T-neighborhood of Θ
anditswavefieldiscontainedinthisneighborhoodfort 0,2T ,uptoharmonicfunctions.
P r s
Proposition2.5. ThecontrolspaceH consistsofCauchydatasupportedoutsideΘwhosewavefieldsare
‹
stationaryharmonicoutsidea2T-neighborhoodofΘatt 0,2T:
“
H h C˜ π h π R h πh 0 . (2.22)
‹ ‹2T ‹2T 2T
“ P ´ “ ´ “ “
(cid:32) ˇ (
ˇ
The second proposition shows that our reflection data (the Cauchy solution operator F, re-
stricted to the exterior of Ω) is determined by the Dirichlet-to-Neumann map, which is the data
usually assumed given in boundary control problems and the inverse problem. As a result, our
methodrequiresnoadditionalinformation,fromatheoreticalstandpoint.
Proposition2.6. Letc1,c2 beL8 wavespeedsonaC1 domainΩ Rn. Extendc1,c2 toΩ‹ Rn Ωby
Ď “ z
settingthemequaltosomec0 C8 Rn .
P p q
DefinesolutionoperatorsF1,F2 correspondingtoc1,c2 asin(2.7),andDirichlet-to-Neumannmaps
2 c2∆ u 0,
t i
pB ´ q “
(cid:12) (cid:12)
Λi: g ÞÑ Bνu(cid:12)R Ω, where $ u(cid:12)R Ω “ g, (2.23)
ˆB ’ (cid:12) (cid:12)ˆB
& u(cid:12)t 0 “ Btu(cid:12)t 0 “ 0.
“ “
IfΛ1 “ Λ2,then F1h(cid:12)(cid:12)RˆΩ‹ “ F2h(cid:12)(cid:12)RˆΩ‹ forallh P H1’%pΩ‹q‘L2pΩ‹q.
2.3 Recoveringinternalenergy
Asadirectapplicationoftheresultsin2.2,weshowhowscatteringcontrolcanrecovertheenergy
of the harmonic almost direct transmission using only data outside Ω, assuming supph Θ Ω.
0
Ă z
IftheNeumannseriesconvergestosomeh C,wecanrecovertheenergydirectlyfromh ,but
8 P 8
if not, Theorem 2.3 allows us to recover the same quantities as a convergent limit involving the
Neumannseries’partialsums.
Proposition 2.7. Let h0 H, T 0, and suppose I π‹Rπ‹R h h0. Then we can recover the
P ą p ´ q 8 “
harmonicalmostdirecttransmission’senergyfromdataobservableonΘ‹ supph0:
Y
ERn hDT ERn h ERn π‹Rh . (2.24)
p q “ 8 ´ 8
` ˘ ` ˘
Wecanalsorecoverthekineticenergyofthealmostdirecttransmission(withnoharmonicextension)from
dataobservableonΘ‹ supph0:
Y
1
KE R h h ,h Rπ Rh Rh . (2.25)
ΘTp T 0q “ 2x 0 0´ ‹ 8´ 8y
9
Proposition 2.8. Let h0 H and T 0, and hk as before. We can recover the energy of the harmonic
P ą
almostdirecttransmissionasaconvergentlimitinvolvingdataobservableonΘ‹ supph0:
Y
E h lim E h E π Rh . (2.26)
DT k ‹ k
p q “ k r p q´ p qs
Ñ8
Similarly,forthekineticenergyofthealmostdirecttransmission,
4KE R h lim E h E h E π Rπ Rh
ΘTp T 0q “ k p kq` p 0q´ p ‹ ‹ kq
Ñ8” (2.27)
2 π Rh , h Rπ Rh 2 h , Rπ Rh Rh .
‹ k k ‹ k 0 ‹ k k
` x ´ y´ x ` y
ı
2.4 Proofs
ProofofTheorem2.2. The proof is mostly a simple application of unique continuation and finite
speedofpropagation.
Equation (2.13) ( ) Let v t,x FR 2TπR2Th be the solution of the wave equation with
ñ p q “ ´ 8
Cauchy data πR h at t 2T. We will often consider Cauchy data at a particular time, and so
2T
8 “
definev v, v .
t
“ p B q
Thedefinitionofh impliesπh h andalso π v 0, 0,since
0 ‹
8 8 “ p qp ¨q “
0 π h π I π R π R h
‹ 0 ‹ ‹ 2T ‹ 2T
“ “ p ´ ´ q 8
π R πR h (2.28)
‹ 2T 2T
“ ´ 8
π v 0, .
‹
“ p qp ¨q
Outside of Θ, then, v 0, and v 2T, are equal to their projections in I, and therefore are
p ¨q p ¨q
stationaryharmonic. Equivalently, v and v arezeroonΘ fort 0,2T.
t tt ‹
B B “
Because c is time-independent, v is also a (distributional) solution to the wave equation. If
t
B
v C R,H1 Rn ,thenLemma2.9appliedto vgives v T, v T, 0onΘ ;itfollows
t t t tt ‹T
B P p p qq B B p ¨q “ B p ¨q “
that v T, is stationary harmonic on Θ . For the general case, choose a sequence of mollifiers
‹T
p ¨q
ρ δ inE R andapplyLemma2.9toρ t v toobtainthesameconclusion.
(cid:15) 1 1(cid:15)
Ñ p q p q˚
Applyingfinitespeedofpropagation(FSP)twice,wefindthatinΘ attimeT,thesolutionv
T
isequaltoh ’swavefield,whichinturnisequaltoh ’swavefield(Figure2.2):
0
8
π v T, π R πR h FSP π R R h π R h FSP π R h def h . (2.29)
T T T 2T T T 2T T T T T 0 DT
p ¨q “ ´ 8 “ ´ 8 “ 8 “ “
However,sincev T, isstationaryharmoniconΘ ,wecanremovetheprojectionontheleft-hand
‹T
p ¨q
side: π R πR h R πR h . Thisprovestheforwarddirectionof(2.13). Moregenerally,
T T 2T T 2T
´ 8 “ ´ 8
itfollowsthatπ R h v T s, R h fors 0,T . Indeed,v T s, R h on
T s T s s DT T s
´ ` 8 “ p ` ¨q “ P r s p ` ¨q “ ` 8
Θ byfinitespeedofpropagation,andusingLemma2.9asaboveimpliesv T s, isstationary
T s
´ p ` ¨q
harmoniconΘ fors 0,T .
‹T s
P r s
´
Equation (2.14) As above, apply Lemma 2.9 to tv. This implies that tv 0,2T Θ 0. Hence v
B B |r sˆ ‹ “
is constant in time in Θ . At time T, we have v T, π R h , and the pressure field v T, is
‹ T T 0
p ¨q “ p ¨q
the harmonic extension of the first component of R h . At time 2T, v equals πR h on Θ
T 0|BΘT 2T 8 ‹
byconstruction,proving(2.14).
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